Lecture Notes on:

Gas Dynamics (EME415)

For EME Students

Adel A. Abdel-Rahman Mech. Eng. Dept.,

Alexandria University

2012/2013

1

Contents

Topic

Page

(1) Introduction & Concepts to Compressible Flow

1

(2) Isentropic Flow

17

(3) Normal Shock Waves

32

(4) Supersonic Wind Tunnels

48

(5) Supersonic Inlets

58

(6) Flow in Ducts with Frictional Effects

70

(7) Propulsion Engine Systems

90

(8) Worked Problems

96

2

References

1. A. H. Shapiro (1953): The dynamics and Thermodynamics of Compressible Fluid Flow, Volume I., New York: Ronald Press.

2. P. H. Oosthuizen and W. E. Carscallen (1997): Compressible Fluid Flow. McGraw-Hill International Edition.

3. B. R. Munson, D. F. Young and T. H. Okishi (1994): Fundamentals of Fluid Mechanics (Chapter 11), 2nd edition, John Wiley & Sons, Inc.

4. B. K. Hodge and K. Koenig (1995): Compressible Fluid

Dynamics with Personal Computer Applications, New Jersy: Prentice Hall.

5. M. A. Saad: Compressible Fluid Flow, Englewood Cliffs, New

Jersey: Prentice Hall.

3

Introduction

Gas Dynamics: Dynamics of Compressible Fluid Flow

???

Density Changes are Large ( > 5%)

Example Applications: Aircraft - Gas pipeline at high pressure

& temperature - Compressors and many others.

A Fluid: Is a substance that deforms CONTINUOUSLY under the

application of a shear stress (what about a solid ??)

Scope of Fluid Mechanics: Fluid Mechanics is concerned with

the study of any system in which a fluid is used such as;

1) Aircraft

2) Automobiles

3) Submarines

4) Rockets

5) Flow around buildings (sky scrapers ??.)

6) Shopping malls

7) All kinds of sports

8) Fluid machines; pumps, compressors, fans

9) Pipelines

10) Medical .. heart , respiratory system,

4

If Fluid is compressible, it is Gas Dynamics

Basic Equations:

Analysis of any system of compressible flow starts with the basic

laws of fluid motion;

1) Conservation of mass

2) Newton's 2nd law of motion

3) Principle of angular momentum

4) 1st law of thermodynamics

5) 2nd law of thermodynamics

In addition, - Equation of state of a perfect gas

- Relation between shear stress and rate of

deformation of a fluid

- Fourier law of heat conduction

Concept of a Continuum:

Matter

Microscopic Macroscopic

Molecular structure

Trace each individual molecule &

set equations for

(Kinetic theory or statistical

mechanics)

In engineering life, interest is in

gross behavior as a continuous

material

Continuum

5

Methods of Analysis:

Define the system to be solved; System or Control Volume

System: it is a fixed identifiable quantity of mass: boundaries

(fixed or movable) surrounding Lagrangian Motion

C.V.: it is an arbitrary volume in space through which fluid

flows: C.V., boundaries (C.S.; fixed, movable, real, imaginary,

at rest, or in motion) Eulerian Motion

Reynolds Transport Theory

It is the technique used to reformulate the system analysis to the

control-volume analysis

i.e; we need to relate the time derivative of a property of a system to

the rate of change of that property within a certain region (C.V.)

If B is any property of the fluid like mass, energy, momentum,

(note that all are extensive properties).

intensive value for a small portion of the fluid is :

dm

dB

the total amount of B in the control volume is BCV, where:

Vd dm

dBB

CV

CV

RTT wants to relate dt

dB with

dt

dBCVsys

Vd dm

dB

dt

d

dt

dB ;e.i

CV

sys

6

Now, if we let:

(1) B = m, we get mass conservation equation;

Since the mass within a fixed-mass system must necessarily be

conserved(dmsys/dt = 0), then:

(2) B = mV, we get linear momentum equation;

And since Newton's second law of motion states:

Then, linear momentum equation for a control volume is:

(3) B = E, we get Energy equation;

CSCV

sys)A.dV( Vd

tdt

dm

CSCV

)A.dV( V Vd Vtdt

)Vm(d

dm

dEe , )A.dV( e Vd e

tdt

dE

CSCV

CSCV

sys)A.dV(

dm

dB Vd

dm

dB

tdt

dB

0.0)A.dV( Vd t

CSCV

dt

)Vm(dF

CSCV

)A.dV( V Vd Vt

F

7

Example:

Water is being added to a storage tank at a rate of 200 liters/min. At the

same time, water flows out the bottom through a 5 cm inside diameter

pipe, with an average velocity of 18 m/s. The storage tank has an inside

diameter of 3 m. Find the rate at which the water level rises or falls.

Solution:

Mass conservation equation is

The water level is falling by 4.5 mm/s

CSCV

)A.dV( Vd t

0.0

0.0AVAVdt

Vd iiweeww

eeieeii AVQAVAVdt

Vd

mm/s 5.4 s/m0045.0A

AVQ

dt

dh

dt

dhA

dt

Vd

hAV ,

T

eei

T

T

h AT

V

i

e

Ve=18m/s

Qi=200lit/mi

n18m/s

8

Example:

The figure shown below is a schematic of a rocket engine mounted on a

test stand in standard atmospheric conditions. The area of the nozzle exit

plane is 225 cm2, the velocity of exhaust gases is 1780 m/s and the mass

flow rate is 1 kg/s. If the pressure at the nozzle exit plane is 180 kPa, find

the thrust force of the rocket engine. Assume steady state, and uniform

(average or one-dimensional) flow conditions at the exit plane.

Solution:

Considering the rocket as a control volume, the linear momentum

equation is:

This equation for steady state (where the rate of change of momentum

within the control volume is zero) and average values for velocities and

densities simplifies to:

CSCV

)A.dV( V Vd Vt

F

Exhaust gases

Rocket

Thrust force (F)

Test stand

Ve=1780 m/s

Ae = 225 cm2

Pe = 180 kPa

e

F

9

i.e; the thrust of the rocket engine is equal to 3580 N to the right

direction, opposite to what is shown in the previous figure.

ie A) (V VA) (V V 0.0 F

0.0V , )VV(m A)p(p - F iieeae

eeae VmA)p(p - F

N 3580F

1780110

22510)100(180 - F

4

3

11

Review of Perfect Gases

For air:

Internal energy, enthalpy and entropy of a perfect gas may be

represented by the following equations:

1k

kRc : pressureconstant at heat Specific

,1k

Rc:olumeconstant vat heat Specific

p

v

1k

k

1

2

1

2

1

2

1

2p1221

1

2

1

2p12pv

T

T

p

p

: toleads p

pRln

T

Tlncssequation ),s(s flow isentropicFor

p

pRln

T

Tlncss , dTcdh , dTcdu

weightmolecular gas theis M

&constant gas theis R

K, J/kg 8314 Ks/m 8314 constant, gas universal theis R

where, M

RR

22

const. c

ck :ratioheat Specific

, const.c-c R :constant Gas

, RT

p :law gasPerfect

v

p

vp

Ks/m 1005c & Ks/m 718c

,m2/s2K 287 = R , 1.4 =k 28.97, = M

22p

22v

11

Speed of Sound ?

Conservation of mass for CV is:

(1)

(2)

Equations (1) and (2) lead to:

, a is called the speed of sound

but what about subscript s . ???

CV

)VC(A)(AC

:flow ldimensiona- one for mass of onConservati

S

22

2

paC

0 as and ,

1p

C

VCP

)CVC(ACA)PP(PA

)VV(mF

:flow ldimensiona-onefor equation Momentum

inout

0.0)A.V( Vd t

CSCV

C

c

x

V

Moving pulse TT

PP

V=0.0

C-V

T

P

C

Stationary pulse TT

PP

T

P

12

Speed of Sound of a perfect gas & isentropic process

For air (where k = 1.4 and R = 287 J/kg K)

m/s T05.20 T (287) (1.4) a

= 340.26 m/s for T=15C (288 K)

Speed of Sound of liquids & solids

For liquids and solids, the bulk modulus of the material is defined as:

a

d

dp

d

dp

Vd

dpV -

For water, bulk modulus of elasticity = 2x109 N/m

2 at 15C.

s/m141410

102 a

3

9

, which is around 4 times the speed of

sound in air at the same temperature.

At same temperature, sound travels through steel at 6000

m/s, which is around 4 times the speed of sound in water.

pk

p

dk

p

dp

process isentropicfor constp

S

k

kRTp

kp

a

gas,perfect afor ,

S

13

Pressure Field Created by a Moving Point Disturbance

Subsonic (u < a)

Motion of

Point source

Airplane flying slower than the speed of sound with

pressure waves moving out from around it

3at

2at

at

a b c d

ut ut ut

14

Supersonic (u > a)

Mach Cone

Zone of Silence

Zone of

action

Airplane flying at supersonic speed with shock waves

moving away and behind the airplane

3at

2at

at

a b c d

ut ut ut

angleMach thecalled is ,

M

1

u

a

ut3

at3Sin

15

Incompressible (u 0.0)

Mach angel () = ??

3at

2at

at

16

Sonic (u = a)

Mach angel () = 90o

Airplane flying at the speed of sound with pressure waves

building up at the airplanes nose to form a shock wave

3at

2at

at

a b c d

ut ut ut